Sect. 1.2 Operations & Properties of Real Numbers
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Transcript of Sect. 1.2 Operations & Properties of Real Numbers
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Sect. 1.2Operations & Properties of Real Numbers Absolute Value Inequalities Addition, Subtraction, Opposites Multiplication, Division, Reciprocals Laws: Commutative, Associative, Distributive
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Absolute Value – Concept Positive distance from 0 on a number line
How far is the number 6 from the 0 point? 6 units
How far is -4.75 from the 0 point? 4.75 units
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Absolute Value – Notation
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Inequalities – Concept Use the number line to understand
< less than > more than
Express the relation of -4 to -7 Either -7 < -4 or -4 > -7 both ways are correct
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Addition of any Two Numbers
14311 64)13(51
02323 8146
13)37(24 2.6)2.6(0
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Opposites (Additive Inverses)
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Opposites and Absolute Value Opposites are a pair of numbers that sum to 0
11.2 and -11.2 -3/17 and 3/17 0 and 0 (zero is it’s own opposite)
The Absolute Value of a number is the positive value of it’s pair of opposites |11.2| = 11.2 and |-11.2| = 11.2 |-3/17| = 3/17 and |3/17| = 3/17
Absolute value brackets can hold expressions |3 – 5| = |-2| = 2 |5 – 3| = |2| = 2 -|22| = -22 and -|-22| = -22
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Multiplication & Division (+ or – ?)
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Fractions with One Negative Sign
ba
ba
ba
ba
Similarly
:These are the Simplest Forms
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The Law of ReciprocalsWhat is 4’s reciprocal?
¼ because 4 (¼) = 1
What is 3½ ’s reciprocal?
2/7 Because (7/2)(2/7) = 1
a and 1/a are Multiplicative Inverses
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Division by Zero
03x
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Another look at Grouping Symbols and the Order of Operations
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Equivalent Expressions Two expressions are Equivalent when they have the
same values for all possible replacements Are the following two expressions equivalent?
3x + 4 and x – 3 + 2x + 7 Yes –
when simplified, the 2nd expression matches the 1st
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I Must Remember: Commutative An operation is Commutative when
Two values can switch positions, and the result is the same.
COMMUTATIVE
Addition and Multiplication are Commutative 6 + 11 = 17 7(9) = 63 11 + 6 = 17 9(7) = 63
Are Subtraction or Division Commutative ?
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Commutative Laws
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I Must Remember: Associative Three values can be computed in different order and
the result is the same. (same operations)
ASSOCIATIVE ASSOCIATIVE
Addition and Multiplication are Associative (19 + 4) + 6 = 23 + 6 = 29 (3)(7)(5) = 21(5) = 105 19 + (4 + 6) = 19 + 10 = 29 (3)(7)(5) = 3(35) = 105
Are Subtraction or Division Associative ?
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The Associative Laws
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The Distributive Law
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)4(3)6(3)46(3
1510)15(10
)3(5)2(5)32(5
xxxx
and (a + b)c = ac + bc
yy
yyyyy
23
)1()3()13(
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Let’s play Name That Law! x + 5 + y = x + y + 5
Commutative … COM 3a + 6 = 3(a + 2)
Distributive … DIST 7x(1 / x) = 7
Reciprocals … RECIP (x + 5) + y = x + (5 + y)
Associative … ASSOC 4(a + 2b) = 8b + 4a
COM, then DIST or DIST, then COM
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Let’s look at next time … Section 1.3 Solving Equations
If we have time today, let’s reviewReducing Numeric Fractions